The behaviour of excitable systems can often be captured with a simpler threshold description. The Integrate-and-Fire model of a neuron is a great example as is the Fire-Diffuse-Fire model of calcium wave propagation in cardiac cells. The project will use tools from dynamical systems theory (bifurcation theory, nonsmooth vector fields, scientific computation) and stochastic processes (Markov chains, Wiener processes) to analyse models with threshold noise. Beginning with studies of single units the project will build up to understand the collective behaviour of interacting threshold units, with applications in neuroscience (rhythm generation, neural computation) and cardiac dynamics (wave propagation, coherent oscillations and arrhythmias).

Cardiac arrhythmia are the leading cause of the death in the UK, killing more people each year than breast cancer, lung cancer and AIDS combined. Among the different kinds of cardiac arrhythmia, atrial fibrillation is the most common one. Here, the smaller chambers of the heart beat so fast that they cannot pump blood anymore, shifting all the blood propelling work of the heart to the larger chambers. This situation is especially problematic under conditions when blood needs to be pumped more quickly (e.g. during exercise) and among the elderly when the heart generally becomes weaker.

One of the precursors of atrial fibrillation are cardiac alternans. Here the heart still exhibits a regular rhythm, but with alternating strength. Only every second heart beat is strong enough to sufficiently contract the heart. A particular form of this arrhythmia are sub-cellular alternans where different parts of a single cell oscillate out of phase. Such a cell does not contract at all, and a group of such cells significantly impairs cardiac contractility. To better understand the emergence of atrial fibrillation and to design treatments, it is vital to gain a comprehensive picture of cardiac alternans.

In this project, we will use a recently developed three dimensional model of an atrial myocyte [2] to investigate the emergence of sub-cellular cardiac alternans. In contrast to earlier work, our approach does not require an ad-hoc compartmentalisation of the cell, but we can work with a realistic representation of the cellular morphology. In turn, this will allow us to better characterise the interaction between different sub-cellular processes that shape cardiac alternans. The challenge is to develop the analysis of alternans for a spatially extended cell model that complements numerical simulations and allows us to predict the onset of alternans more efficiently. Keeping in mind that any drug treatment acts first at the single cell level, our approach will help to identify potential targets for pharmaceutical intervention in cardiac therapy.

A number of fascinating and important biological processes involvevarious kinds of spatial patterns: spatial patterns on animal skins, orthe very regular organ arrangements found in plants (called phyllotaxis)for instance. These patterns often originate at very small scales, andtheir onset can only be seen using very recent microscope and imageanalysis techniques.Among several families of models for biological patterning, one of thesimplest is based on the idea that mobile substances (called morphogens) are acting upstream of their targets, which respond locally to a globallydefined gradient pattern.In this project one will consider models where targets are themselvesmobile morphogens, potentially regulating their own input. One will study the effect of such spatial feedback on patterning. To do so, one will rely on a class of models which are biologically relevant, tractable analytically, and not much studied yet in a context with spatial interactions. A class of models which meet all this criteria is provided by piecewise-linear differential equations.

Mathematical Neuroscience is increasingly being recognised as a powerful tool to complement neurobiology to understand aspects of the human central nervous system. The research activity in our group is concerned with developing a sound mathematical description of sub-cellular processes in synapses and dendritic trees. In particular we are interested in models of dendritic spines [1], which are typically the synaptic contact point for excitatory synapses. Previous work in our group has focused on voltage dynamics of spine-heads [2]. We are now keen to broaden the scope of this work to include developmental models for spine growth and maintenance, as well as models for synaptic plasticity [3]. Aberrations in spine morphology and density are well known to underly certain brain disorders, including Fragile X syndrome (which can lead to attention deficit and developmental delay) and depression [4]. Computational modelling is an ideal method to do in-silico studies of drug treatments for brain disorders, by modelling their action on spine development and plasticity. This is an important complementary tool for drug discovery in an area which is struggling to make headway with classical experimental pharmaceutical tools.

The mathematical tools relevant for this project will be drawn from dynamical systems theory, biophysical modelling, statistical physics, and scientific computation.

Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a non-smooth dynamical system with a threshold [1]. It has recently been shown [2] that one way to model the variability of neuronal firing is to introduce noise at the threshold level. This project will develop the analysis of networks of synaptically coupled noisy neurons. Importantly it will go beyond standard phase oscillator approaches to treat strong coupling and non-Gaussian noise. One of the main mathematical challenges will be to extend the Master-Stability framework for networks of deterministic limit cycle oscillators to the noisy non-smooth case that is relevant to neural modelling. This work will determine the effect of network dynamics and topology on synchronisation, with potential application to psychiatric and neurological disorders. These are increasingly being understood as disruptions of optimal integration of mental processes sub-served by distributed brain networks [3].

Cell signalling effects have crucial roles to play in a vast range of biological processes, such as in controlling the virulence of bacterial infections or in determining the efficacy of treatments of many diseases. Moreover, they operate over a wide range of scales, from subcellular (e.g. in determining how a particular drug affects a specific type of cell) to organ or population (such as through the quorum sensing systems by which many bacteria determine whether or not to become virulent). There is therefore an urgent need to gain greater quantitative understanding of these highly complex systems, which are well-suited to mathematical study. Experience with the study of nonlinear dynamical systems would provide helpful background for such a project.

Whilst the dynamics of the DNA double helix are extremely complicated, a number of well-defined modes of vibration, such as twisting and bending, have been identified. At present the only accurate models of DNA dynamics involve large-scale simulations of molecular dynamics. Such approaches suffer two major drawbacks: they are only able to simulate short strands of DNA and only for extremely short periods (nanoseconds). the aim of this project is to develop simpler models that describe vibrations of the DNA double helix. The resulting systems of equations will be used to simulate the dynamics of longer chains of DNA over long timescales and, hence, allow larger-scale dynamics, such as the unzipping of the double helix, to be studied.

Most human tissues are perfused by an evolving network of blood vessels which supply nutrients to (and remove waste products from) the cells. The growth of this network (via vasculogenesis and angiogenesis) is crucial for normal embryonic and postnatal development, and its maintenance is essential throughout our lives (e.g. wound healing requires the repair of damaged vessels). However, abnormal remodelling of the vasculature is associated with several pathological conditions including diabetic retinopathy, rheumatoid arthritis and tumour growth.

The phenomena underlying tissue vascularisation operate over a wide range of time and length scales. These features include blood flow in the existing vascular network, transport within the tissue of blood-borne nutrients, cell division and death, and the expression by cells of growth factors such as VEGF, a potent angiogenic factor. We have developed a multiscale model framework for studying such systems, based on a hybrid cellular automaton which couples cellular and subcellular dynamics with tissue-level features such as blood flow and the transport of growth factors. This project will extend and specialise our existing model to focus on particular applications in one of the following areas: wound healing, retinal angiogenesis, placental development, and corpus luteum growth. This work would require a significant element of modelling, numerical simulation and computer programming.

Molecular Beam Epitaxy is a process by which single atoms are slowly deposited on a surface. These atoms diffuse around the surface until they collide with a cluster or another atom and become part of a cluster. Clusters remain stationary. The distribution of cluster sizes can be measured, and is observed to exhibit self-similarity. Various systems of equations have been proposed to explain the scaling behaviour observed. The purpose of this project is to analyse the systems of differential equations to verify the scalings laws observed and predict the shape of the size-distribution. The relationship of equations with other models of deposition, such as reactions on catalytic surfaces and polymer adsorption onto DNA, will also be explored.

The random deposition of particles onto a surface is a process which arises in many subject areas, and determining its efficiency in terms of the coverage attained is a difficult problem.

In one-dimension the problem can be viewed as how many cars can be parked along a road of a certain length; this problem is similar to a problem in administering gene therapy in which polymers need to be designed to package and deliver DNA into cells.

Here one wishes to know the coverage obtained when one uses a variety of polymer lengths to bind to strands of DNA.

The project will involve the solution of recurrence relations, and differential equations, by a mixture of asymptotic techniques and stochastic simulations.

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Robustness of biochemical network dynamics with respect to mathematical representation

In the recent years, a lot of multi-disciplinary efforts have beendevoted to improving our understanding of the dynamics of interactionsbetween the many types of molecules present in biological cells. Thishas led to a widespread viewpoint where networks of genes, proteins andother biochemical species are considered at once, as complex dynamicalsystems from which the global state of cells emerge.Several mathematical formalisms are used to represent these systems,from discrete or boolean models to differential equations. One strikingfact, especially regarding models of developmental processes, is that anumber of relevant properties of these networks can be capturedsimilarly by all these formalisms, like for instance the property ofbistability.One possible interpretation of this independence with respect toformalism is that biological regulatory systems are most often extremelyrobust.The project will start by developing parallel models – using differentformalisms – of actual biological networks whose behaviour is known.Elaborating on these examples the theoretical and practical implicationsof this notion of robustness will be explored.

Calcium is critically important for a large number of cellular functions, such as muscle contraction, cardiac electrophysiology, secretion, synaptic plasticity, and adaptation in photoreceptors [1]. Mechanisms by which a cell controls its Calcium concentration are of central interest in cell physiology. The recent use of Calcium specific fluorescent reporter dyes and digital videomicroscopy has begun to reveal the complexity of Calcium dynamics in spatially extended cellular systems. Calcium signalling in a wide diversity of cell types frequently occurs as Calcium oscillations. These do not generally occur uniformly throughout the cell but are initiated at a specific site and spread in the form of waves. The fluorescent imaging of localised elementary Calcium release events has now made it clear that Calcium release is a stochastic process that occurs at spatially discrete sites that are clusters of receptors in the endoplasmic or sarcoplasmic reticulum. Mathematical modelling is an ideal tool for capturing the details of how intracellular Calcium waves spread throughout a cell and subserve physiological and pathological signals, especially in light of current resolution limitations of imaging technologies. In particular the stochastic Fire-Diffuse-Fire (FDF) model [2] is an ideal starting point for the development of a computationally economical framework to allow fast simulations of realistic cell geometries with large numbers of release sites. This project will focus on developing cell models to track waves that allow targeted release of calcium in the nuclear region of eukaryotes. Nuclear oscillations in calcium are especially important as they can drive downstream responses for gene expression. The model will be extended to include calcium decoders (such as a nuclear-localised calcium and calmodulin-dependent protein kinase) and developed to model the symbiotic signalling pathway of legumes during root nodulation [3].

This project will mainly draw from the toolbox of computational cell biology (including GPU programming in Python) to address important open problems in cellular calcium signalling in plants.

Neural field models describe the coarse grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in 2D, where they are well known to generate rich patterns of spatio-temporal activity. Typical patterns include localised solutions in the form of travelling spots as well as spiral waves [1]. These patterns are naturally defined by the interface between low and high states of neural activity. This project will derive the dimensionally reduced equations of motion for such interfaces from the full nonlinear integro-differential equation defining the neural field. Numerical codes for the evolution of the interface will be developed, and embedded in a continuation framework for performing a systematic bifurcation analysis. Weakly nonlinear theory will be developed to understand the scattering of multiple spots that behave as auto-solitons, whilst strong scattering solutions will be investigated using the scattor theory that has previously been developed for multi-component reaction diffusion systems [2].

Calcium oscillations have long been recognised as a main pathway with which cells translate external stimuli into intracellular responses such as enzyme secretion, neurotransmitter production or cell contraction [1]. Over the last years, it has emerged that calcium oscillations often do not occur uniformly across a cell, but that either different parts of a cell oscillate out of phase with respect to each other or that cellular oscillations actually correspond to traveling calcium waves. The importance of space in shaping intracellular calcium oscillations has recently been highlighted by the discovery of the STIM-ORAI machinery [2]. Here, translocation of intracellular molecules (STIM) to designated areas close to the cell membrane (ORAI) are responsible for initiating and maintaining calcium oscillations.

A large body of experimental data convincingly suggests that most of the information of intracellular calcium oscillations is encoded in their frequency and sometimes in their amplitude. The STIM-ORAI system now shows that only if the cellular calcium oscillations occur through STIM and ORAI certain genes are activated. Intracellular calcium oscillations that look identical but involves different molecular partners fail to initiate a genetic response [3].

In this project, we will develop a spatially extended model for STIM-ORAI induced calcium oscillations that will explain the still unknown mechanisms behind the long periods of calcium oscillations. Introducing the pathway that is responsible for gene activation, we will study the signalling cascade that links calcium oscillations to gene expression with a special emphasis on the emergence of calcium microdomains and exchange mechanisms between the cell cytoplasm and nucleus. Given the importance of STIM-ORAI dependent oscillations in cells of the immune system, our work has direct implications to strengthen human health.

The project will involve model design based on published experimental data and mathematical techniques for partial differential equations, delayed differential equations and stochastic processes.

Insects have evolved diverse and delicate morphological structures in order tocapture the inherently low energy of a propagating sound wave. In mosquitoes, thecapture of acoustic energy, and its transduction into neuronal signals, is assistedby the active mechanical participation of actuators called scolopidia.

When a sound wave reaches the head of a mosquito, the antenna oscillates under theaction of the external pressure field (passive component) and of the force provided bythe mechanical actuators (active component). The latter is particularlyrelevant for sexual recognition: when a male mosquito hear the flyby of a female, hisantennal oscillation are greatly amplified by the scolopidia. In other words, theantenna of a male is tuned very sharply around the frequency and intensity of afemale flyby.

Recent studies have shown that mosquitoes of either sex use both their antenna andtheir wing beat to select a partner: understanding how their hearing system workscould help us controlling the population of species that carry viral diseases.

Even though some models of mosquitoes hearing systems have been proposed in the past,a number of key questions remain unanswered. Where do the mechanical actuators gettheir energy? How do they twitch? How is the mechanical motion of the antennatransformed into an electric signal? Do neurones control the mechanical motion? Howdoes the brain of a mosquito process the neural information and distinguish varioussources of sound? Is the sexual recognition entirely based on sound perception, or isit also influenced by olfactory signals? Is the antenna sensitive to sounds fromdifferent directions?

Lung inflammation and airway hyperresponsiveness (AHR) are hallmarks of asthma, but their interrelationship is unclear. Excessive shortening of airway smooth muscle (ASM) in response to bronchoconstrictors is likely an important determinant of AHR. Hypercontractility of ASM could stem from a change in the intrinsic properties of the muscle, or it could be due to extrinsic factors such as chronic exposure of the muscle to inflammatory mediators in the airways with the latter being a possible link between lung inflammation and AHR. The aim of this project will be to investigate the influence of chronic exposure to a contractile agonist on the force-generating capacity of ASM via a cell-level model of an ASM cell. Previous experimental studies have suggested that the muscle adapts to basal tone in response to application of agonist and is able to regain its contractile ability in response to a second stimulus over time. This is thought to be due to a transformation in the cytoskeletal components of the cell enabling it to bear force, thus freeing up subcellular contractile machinery to generate more force. Force adaptation in ASM as a consequence of prolonged exposure to the many spasmogens found in asthmatic airways could be a mechanism contributing to AHR seen in asthma. We will develop and use a cell model in an attempt to either confirm this hypothesis or determine other mechanisms that may give rise to the observed phenomenon of force adaptation.

Computational Cell Biology (CSB) uses techniques from nonlinear dynamical systems, partial differential equations and stochastic processes to gain a deeper insight into the reliability and robustness of cellular signalling cascades. By combining analytical and numerical approaches, CSB plays a major role in the discovery and quantitative descriptions of key biological processes. Mathematical models that are tailored to specific biological questions can yield answers that are still out of reach for cutting-edge experimental approaches.

In the current project we will explore how the spatial arrangement of the molecular machinery affects cellular signal transduction. A key feature of cells is to translate external stimuli into cellular responses. Cells in the pancreas produce insulin when extracellular markers indicate high blood sugar levels, neurons in the brain release chemical messengers to their neighbours upon electrical stimulation, and heart cells contract more effectively when they experience a rush of adrenaline. Cells use advanced molecular machinery to trigger the appropriate reaction for a given stimulus. In recent years, convincing evidence has emerged that cells employ spatio-temporal patterns to achieve this task. For instance, complex oscillations and travelling waves of intracellular calcium have been observed, where the frequency of the oscillations and the spatial spread of the waves tightly correlate with the external stimulation.

Around 25% of the 50million epilepsy sufferers worldwide are not responsive to antiepileptic medication; improved understanding of this disorder has the potential to improve diagnosis, treatment and patient outcomes. The idea of modelling the brain as a complex network is now well established. However, the emergence of pathological brain states via the interaction of large interconnected neuronal populations remains poorly understood. Current theoretical study of epileptic seizures is flawed by dynamical simulation on inadequate network models, and by the absence of customised network measures that capture pathological connectivity patterns.

This project aims to address these deficiencies via improved computational models with which to investigate thoroughly the influence of the geometry and connectivity of the human brain on epileptic seizure progression and initiation, and the development of novel network measures with which to characterise epileptic brains. Such investigations will be informed by exhaustive patient datasets (such as recordings of neural activity in epilepsy patients and age-matched controls), and will be used to study (i) improved diagnostic strategies, (ii) the influence of treatment strategies on seizure progression and initiation, and (iii) the identification of key sites of epilepsy initiation.

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A systems biology investigation of pathogen-host homeostasis and disease modulation

The development of antibiotics has been one of the great successes in modern medicine, providing a step-change in public health. However, many bacteria have now developed antibiotic resistance; this is recognised as a global healthcare threat, and chosen as a G8 Global Challenge and 2014 Longitude Prize topic. Long-term solutions require novel strategies going beyond traditional drug development, especially given the failure to discover and develop new drugs. Recent work reveals a novel signalling mechanism underlying patho-adaptation of certain bacteria. A quantitative understanding of this mechanism provides the potential to aid the development of entirely new treatments for bacterial infection.

This project will employ a Systems Biology approach: detailed network models will be used to analyse relevant gene regulation networks and signalling mechanisms, controlling bacterial patho-adaptation and host immune responses; subsequently, multiscale models will be developed to study the multicellular system, incorporating inter- and sub-cellular detail. Comprehensive in silico investigation will be complemented by powerful multiscale asymptotic methods for model reduction and analysis, which provide more intuitive and tractable descriptions. Ongoing experimental work led by Dr Jafar Mahdavi (Life Sciences) will be central to this research: biological experiments will guide model-building and parameterisation, leading to new experimentally-testable model-driven hypotheses.

One of the greatest challenges of biology is to decipher the relation between genotype and phenotype. One core difficulty in this task is that this relation is not a map; the proteins whichare produced thanks to the information contained in the genome are themselves used to control which parts of the genome are being used in a given situation. To understand the effectof this feedback between genes and their product it is crucial to consider the dynamics of this process.The term 'gene network' refers to a set of genes which regulate each other; understanding the dynamics of gene networks is thus crucial to decipher the genotype-phenotype relation. Mathematical models of gene networks have been proposed since the 1960's, among which the class of Boolean models has proved very successful. Because of the discrete nature of these models, the effect of time is often described using representations inspired by manufactured computing device, where the genes are updated in parallel, or in series. However, the updating scheme of genes could in principle be much more general. In this project, one will investigate the effect of such a generalization. One will consider arbitrary update schemes, both deterministic and stochastic, notably in relation to the dynamics of continuous models of gene networks.

A very general phenomenon is the fact that coupled oscillators tend to naturally synchronize [1]. This simple fact takes many forms observable in real life: synchronization of applause after a concert, of neural cells, of flashing fireflies, and many other. A complete understanding of this phenomenon, depending on the particular dynamics of individual oscillators or the nature of their coupling, is still an on-going topic of mathematical research.However general, the synchronization of coupled oscillators is not a universal rule. In this project, one will study a situation where it indeed does not seem to occur: the division cycles of cells in a growing tissue does not seem to be synchronized, as observed in recent data form plant tissues. A plausible explanation is that the divisions of cell induce a change in the coupling between cells, which is mostly due to physical or chemical exchanges between neighbouring cells. Relying on simplified representations, one will consider the effect of growth, whereby the coupling structure of a system changes in time, on synchronization in populations of oscillators.

Airway remodelling in asthma has until recently been associated almost exclusively with inflammation over long time-scales. Current experimental evidence suggests that broncho-constriction (as a result of airway smooth muscle contraction) itself triggers activation of pro-remodelling growth factors that causes airway smooth muscle growth over much shorter time-scales. This project will involve the coupling of sub-cellular mechano-transduction signalling pathways to biomechanical models of airway smooth muscle cells and extra-cellular matrix proteins with the aim of developing a tissue-level biomechanical description of the resultant growth in airway smooth muscle.

The mechano-transduction pathways and biomechanics of airway smooth muscle contraction are extremely complex. The cytoskeleton and contractile machinery within the cell and ECM proteins surrounding it are thought to rearrange dynamically (order of seconds). The cell is thought to adapt its length (over 10s of seconds). To account for all these processes from the bottom-up and generate a tissue level description of biological growth will require the combination of agent-based models to biomechanical models governed by PDEs. The challenge will be to come up with suitably reduced models with elegant mathematical descriptions that are still able to reproduce observed experimental data on cell and tissue scales, as well as the different time-scales present.

While this study will be aimed specifically at airway remodelling, the methodology developed will have application in multi-scale models of vascular remodelling and tissue growth in artificially engineered tissues. Initially models will be informed by data from on-going experiments in Dr Amanda Tatler's lab in Respiratory Medicine but there will also be the opportunity to design new experiments based on model results.

Macrophages are a type of white blood cell, a vital component of the immune system, and play a complex role in tumour growth and other diseases. Macrophage precursors, called monocytes, are produced in the bone marrow and enter the blood, before leaving the bloodstream (extravasating). Monocyte extravasation requires adhesion to, and active movement through, the blood vessel wall, both of which are highly regulated processes. Once in the tissue, monocytes begin to differentiate into macrophages, and it has become clear that the tissue micro-environment is a crucial determinant of macrophage function [1]. A spectrum of phenotypes have been identified: at one end, macrophages produce a variety of signals that are beneficial to a tumour, including those that promote the formation of new blood vessels and suppress inflammation. At the other end of the scale, inflammation is promoted and appropriately stimulated macrophages can kill tumour cells.

This project will consider in some detail the mechanisms that regulate monocyte extravasation and macrophage phenotype selection. Initially, mathematical models will be formulated as systems of ordinary differential equations describing transitions between monocyte subpopulations (for example, those fully adherent to the vessel wall, and those that are actively moving through the wall), regulated by various signalling and adhesion molecules. Research on phenotype selection will determine whether the dynamics can be manipulated by subsequent external intervention. For example, if the system is bistable, it may be possible to force a switch from a deleterious to a beneficial phenotype. Relevant signal transduction pathways will be modelled in detail, and the law of mass-action will be used to derive systems of ODEs. Where possible, model reductions based on a separation of timescales will be used to simplify the system, and analytical and numerical approaches will be used to characterise steady state structure and bifurcations as tissue conditions vary.

Inference of the world around us is made by processing sensory signals in the brain and relating them to memories of previous experience. The study of this process has generated a number of candidate frameworks, with perhaps the most popular being ‘Bayesian cognition'. This powerful statistical description posits that the central nervous system of animals is capable of integrating prior probabilities with new sensory data in an optimal way to make perceptual decisions1. How this process could be realised in dynamic circuits of neurons is as yet unclear2. Additionally, while there are some spectacular experimental data on the capability of humans to accomplish such Bayes-optimal computations, strong evidence only comes from a limited set of experiments3, other evidence often assuming Bayesian algorithms a priori4 and there is an influential literature demonstrating the failures of human information processing to incorporate prior probabilities5. This project will probe the mechanisms involved in perceptual inference in a multi-disciplinary way mixing techniques from cognitive psychology and neuroimaging with those from mathematical neuroscience.

A PhD student will approach this question on two frontiers: Phenomenologically, they will use advanced psychophysical techniques and mathematical modelling 6 to study how humans integrate prior probabilities and sensory information more generally (i.e. investigate the parameter space where the Bayesian observer model applies) and measure neuronal activity during such processes noninvasively7. Computationally, they will then investigate a new class of forward models for the generation of brain rhythms based on mean-field reductions of synaptically interacting nonlinear integrate-and-fire systems8. These are ideally suited for studying the phase-amplitude coupling of brain rhythms in hierarchical cortical networks that have been reported in human and animal studies on the integration of prior information9,10. The implementation of a Bayesian machine in this architecture will shed light on the debate about how higher frequency 'gamma oscillations' can communicate sensory feed-forward information, whilst top-down feedback is mediated by lower frequency 'alpha-' or 'beta-' band oscillations11. The PhD student will be actively involved not only in model development and delivery but the acquisition of psychophysical data. Initial experiments will be used to validate the model, progressing through to the design of new experiments to test model-generated hypotheses about the breakdown of computation in mental illness. In particular the project will explore the role of NMDA vs. AMPA glutamatergic receptors in subserving neurodynamics for Bayesian cognition and their disturbance in schizophrenia12.

The project will be jointly supervised by Dr Markus Bauer (School of Psychology) and Professor Stephen Coombes (School of Mathematical Sciences).

One of the holy grails of the theoretical neuroscience community is to develop a tractable model of neural tissue. This must necessarily involve a single cell model, capable of generating spikes of activity (so-called action-potentials), that when connected into a synaptic network can generate the rich repertoire of behaviour seen in a real nervous system. For all of the popular conductance-based single neurons models, and also the simpler integrate-and-fire variety, the understanding of network dynamics has proved elusive. In essence this is because we have not yet developed an appropriate mathematical framework to understand the neurodynamics of spiking networks. To date progress in this area has been restricted to firing rate neural models, which cannot adequately capture known spike-train correlations. Interestingly, the recently proposed Lighthouse model of Hermann Haken is a candidate single neuron model that may allow a bridge to be built between spiking neuron models and firing rate descriptions. Indeed in the limit of slow synaptic interactions it may be shown to reduce to the oft-studied Amari firing rate model. Importantly the Lighthouse model is sufficiently simple that it may also be analysed at the network level, even for fast synaptic responses. Hence, a comprehensive study of a network of synaptically coupled Lighthouse neurons may pave the way for the development of a specific exactly soluble neurodynamics. This may also shed light on how best to develop a more general approach valid for more detailed models of coupled spiking neurons. This project will pursue the study of the Lighthouse network using techniques from dynamical systems theory and statistical physics, building upon emerging techniques and principles from the physics of complex systems. As it will closely focus on the generation of realistic spike-train correlations from a mathematical model it will benefit enormously from locally available multi-electrode array data collected from both in-vitro and in-vivo neuronal ensembles.

Shear induced chaos has recently been shown to be an important mechanism for determing the response of conductance based models of single neurons to time-dependent (typically periodic) input [i]. This Phd project will develop a natural phase-amplitude coordinate system [ii] for describing reduced networks of synaptically interacting neurons. Network states, including phase-locking, synchrony, heteroclinic cycles, and routes to chaos, will be analysed using techniques from dynamical systems theory (both analytical and numerical) to understand fundamental aspects of information processing within the central nervous system including network reliability in the presence of shear.

There is increasing evidence to suggest that chronic pain is a disease that can alter brain function. In particular neuroimaging studies have demonstrated structural remapping and functional reorganisation of brain circuits under various pain conditions. In parallel, preclinical models have demonstrated that chronic pain causes long-term neuroplasticity. For a recent review see [1].

In theory, physiological changes at the single-unit, multi-unit, and circuitry levels can be used as predictors of pain, and ultimately to guide targeted neuromodulation of specific brain regions for therapeutic purposes. The Pain Imaging group at Nottingham is developing circuit level imaging biomarkers (using MRI techniques) to track such physiological changes. The complementary statistical techniques for prediction (and identification of brain states associated with pain) and computational modelling that would allow in-silico design of pain therapies are skill sets that exist within the School of Mathematical Sciences. Thus Nottingham is well positioned to develop multidisciplinary research into the mechanisms of pain-related phenomena in the brain that can offer insights into novel approaches for the diagnosis, monitoring, and management of persistent pain.

Aims and objectives

In light of recent breakthroughs in the statistical analysis of brain network signals [2] and computational models of interacting neuronal populations [3], as well as locally available data sets from the Pain Imaging group, our aim is to equip a PhD student with multi-disciplinary skills for understanding how humans experience pain. The objective is for them to develop a novel systems perspective of pain as a complex multidimensional experience that can be understood with the modern tools of applied mathematics and statistics.

Although activation patterns may vary, the regions most consistently reported to have increased blood-oxygen-level-dependent signals associated with experimentally induced pain include the thalamus, somatosensory cortex, anterior cingulate cortex, prefrontal cortex, insula, and the cerebellum, forming a so-called pain matrix. We will develop network models of this system of interacting neural populations building on recent work in [3]. This will allow us to explore the mechanisms for the emergence of functional connectivity associated with normal activation of the ‘pain matrix’, and dysfunctional connectivity associated with the experience of chronic pain. The transition between the two states will be studied, with a particular focus on the dependence of the functional connectivity patterns on the dynamics of a sub-population, the dynamics of synaptic currents, and plasticity of interconnections (and of course disturbances in each, mimicking various forms of sensitisation, channelopathies, sub-circuit over-activation, etc.). The development of an in silicomodel will also allow the design of restorative stimulation protocols, such as via deep brain stimulation or patient-controlled real-time feedback, to alleviate pain. The mathematical challenge will be to understand how a dysfunctional `pain matrix' state induced within the model environment can be coaxed back to a normal activation pattern.

Statistical methods will be developed to decode neuroimaging signals and predict a sensory pain experience on the basis of spatially correlated fMRI voxels. Exponential random graph models (ERGMs) will allow us to gain deeper insights into the complex neurobiological interactions and changes that occur in many disorders. Although ERGMs have been extensively utilised in social science to analyse highly complex networks, it is only until recently that they have been successfully used to study brain networks using resting fMRI data showing some very promising results [2].

Training

The student will do a laboratory rotation in the Pain Imaging group, to appreciate the data sets that are available to work with. Training on Neuroimaging data acquisition and analysis will be provided by participation at the MSc Translatianal Neuroimaging (Course director: D Auer)

The student will learn about advanced techniques in Computational Neuroscience by attending the course G14TNS Theoretical Neuroscience (School of Mathematical Sciences). The student will also learn about advanced statistical computational techniques such as Markov Chain Monte Carlo (MCMC) by attending the course G14CST and courses from the Academy for PhD Training in Statistics (APTS).

A human heart beats more than a billion times during the average lifespan, and is required to do so with great fidelity. The ventricular (larger) chambers of the heart are responsible for generating the force that propels blood to the lungs and body [1]. Under sedentary conditions, the atrial (smaller) chambers make only a minor contribution to blood pumping. However, during periods of increased hemodynamic demand, such as exercise, atrial contraction increases to enhance the amount of blood within the ventricles before they contract. This `atrial kick' is believed to account for up to 30% extra blood pumping capacity. Deterioration of atrial myocytes, i.e. muscle cells, with ageing causes the loss of this blood pumping reserve, thereby increasing frailty in the elderly. Atrial kick is also lost during atrial fibrillation (AF), the most common form of cardiac arrhythmia. The stagnation of blood within the atrial chambers during AF can cause thrombus formation, leading to thromboembolism. Approximately 15% of all strokes occur in people with AF. As shown in numerous reports, the genesis and maintenance of AF is causally linked to the dysregulation of calcium signalling, which is bidirectionally coupled to the membrane potential of the cell [2-4].

In this project, we will investigate how changes in the membrane potential lead to changes in the intracellular calcium concentration, which in turn feeds back to the temporal evolution of the membrane potential. We will employ a recently developed three-dimensional model of an atrial myocyte with a biologically realistic distribution of calcium release sites. Through detailed numerical simulations we will achieve a better understanding of how the specific morphology of atrial myocytes impacts on the membrane driven generation of calcium transients, and how clinically relevant pathologies like early after depolarisation and delayed after depolarisation are shaped through the interaction of localised calcium transients near the plasma membrane and the membrane potential itself.

Perception is, in general, an active process. Animals use purposive control of their sense organs (e.g., eyes, fingers, whiskers) to obtain sensory information to guide current behavioural goals (‘active sensation’). Rodents actively move use whiskers to sense the world around them and, since these whisker movements can be precisely imaged in an awake, behaving animal, it is an ideal system in which to investigate active sensing. Stereotypical patterns of 'whisking' allow the animal to transduce mechanical forces arising upon impact of their whiskers with objects, textures and surfaces, into patterns of spiking activity that their brain uses to form a representation of the real world [1]. However, surprisingly little is known about how the mechanics of the whisker and its dynamics upon impact can lead to a behaviourally useful tactile sensation.

This project will develop models of rodent whiskers, based upon elastic conical beam mechanics [2], and use the modern tools of applied mathematics to understand their behaviour when brushed and tapped against objects. In particular the project will develop continuum and finite-element models coupled to nonsmooth event driven dynamics to describe collisions, such as those that characterise texture exploration with a "stick-slip-ring" pattern of activity [3]. This computational model will be validated against high speed video of mice moving their whiskers in both free-air and collision whisking, provided by Dr Rasmus Petersen, Faculty of Life Sciences, University of Manchester.

The dynamics of the collision events will be analysed mathematically using impact oscillator theory, with a natural extension of these ideas to account for the spatial extent of whiskers. The work will then progress to models of mechano-transduction, using force estimates at the whisker follicle to understand how these drive mechano-receptors to generate action-potentials, and ultimately how the rodent brain can develop a neural representation of the external world.

Partitioned-domain concurrent multiscale modelling(Or- How does one get cheap, but accurate, models?)

Multiscale modeling is an active area of research in all scientific disciplines. The main aim is to address problems involving phenomena at disparate length and/or time scales that span several orders of magnitude! An important multiscale-modeling type is known as partitioned-domain concurrent modelling. This type addresses problems that require a fine-scale model in only a small part of the domain, while a coarse model is employed in the remainder of the domain. By doing this, significant computational savings are obtained compared to a full fine-scale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem.

Challenges for students: * How can one couple, e.g., discrete (particle) systems with continuum (PDE) models? * Or a fine-scale PDE with a coarse-scale PDE?* How can one decide on the size and location of the fine-scale domain?* Is it possible to proof the numerically observed efficiency of concurrent multiscale models? * Can the multiscale methodology be applied to biological growth phenomena (e.g., tumours) where one couples cell-based (agent-based) models with continuum PDE models?

Depending on the interest of the student, several of these issues (or others) can be addressed.Also, the student is encouraged to suggest a second supervisor, possibly from another group!

The activity of neurons within the brain can be detected by function magnetic resonance imaging (fMRI) and magnetoencephalography (MEG). The techniques record observations up to 1000 times a second on a 3D grid of points separated by 1-10 millimetres. The data is therefore high-dimensional and highly correlated in space and time. The challenge is to infer the location, direction and strength of significant underlying brain activity amongst confounding effects from movement and background noise levels. Further, we need to identify neural activity that are statistically significant across individuals which is problematic because the number of subjects tested in neuroimaging studies is typically quite small and the inter-subject variability in anatomical and functional brain structures is quite large.

Many forms of lung disease are characterised by excess fibrous tissue developing in the lungs. Fibrosis is currently diagnosed by human inspection of CT scans of the affected lung regions. This project will develop statistical techniques for objectively assessing the presence and extent of lung fibrosis, with the aim of identifying key factors which determine long-term prognosis. The project will involve developing statistical models of lung shape, to perform object recognition, and lung texture, to classify healthy and abnormal tissue. Clinical support and data for this project will be provided by the School of Community Health Sciences.

The spread of so-called superbugs such as MRSA within healthcare settings provides one of the major challenges to patient welfare within the UK. However, many basic questions regarding the transmission and control of such pathogens remain unanswered. This project involves stochastic modelling and data analysis using highly detailed data sets from studies carried out in hospital, addressing issues such as the effectiveness of patient isolation, the impact of different antibiotics, and the way in which different strains interact with each other.

When new infections emerge in populations (e.g. SARS; new strains of influenza), no vaccine is available and other control measures must be adopted. This project is concerned with addressing questions of interest in this context, e.g. What are the most effective control measures? How can they be assessed? The project involves the development and analysis of new classes of stochastic models, including intervention models, appropriate for the early stages of an emerging disease.

The structure of the underlying population usually has a considerable impact on the spread of the disease in question. In recent years the Nottingham group has given particular attention to this issue by developing, analysing and using various models appropriate for certain kinds of diseases. For example, considerable progress has been made in the understanding of epidemics that are propogated among populations made up of households, in which individuals are typcially more likely to pass on a disease to those in their household than those elsewhere. Other examples of structured populations include those with spatial features (e.g. farm animals placed in pens; school children in classrooms; trees planted in certain configurations), and those with random social structure (e.g. using random graphs to describe an individual's contacts). Projects in this area are concerned with novel advances in the area, including developing and analysing appropriate new models, and methods for statistical inference (e.g. using pseudo-likelihood and Markov chain Monte Carlo methods).

Data-analysis for real-life epidemics offers many challenges; one of the key issues is that infectious disease data are usually only partially observed. For example, although numbers of cases of a disease may be available, the actual pattern of spread between individuals is rarely known. This project is concerned with the development and application of methods for dealing with these problems, and involves using Markov Chain Monte Carlo (MCMC) techniques.

During the last decade there has been a significant progress in the area of parameter estimation for stochastic epidemic models. However, far less attention has been given to the issue of model adequacy and assessment, i.e. the question of how well a model fits the data. This project is concerned with the development of methods to assess the goodness-of-fit of epidemic models to data.

There has been considerable interest recently in models for epidemics on networks describing social contacts. In these models one first constructs an undirected random graph, which gives the network of possible contacts, and then spreads a stochastic epidemic on that network. Topics of interest include: modelling clustering and degree correlation in the network and analysing their effect on disease dynamics; development and analysis of vaccination strategies, including contact tracing; and the effect of also allowing for casual contacts, i.e. between individuals unconnected in the network. Projects in this area will address some or all of these issues.